# The Mach-Zehnder interferometer explained

A wave can experience a phase shift upon full or partial reflection. This is a feature that is applied in the Mach-Zehnder interferometer to induce phase differences in split light waves en thus invoke interference effects when they are combined again. In the Mach-Zehnder interferometer beam splitters are used to achieve several things:

• split a light beam in two equal and synchronous beams, a reflected beam and a strait-through passing beam,
• induce a phase shift in the reflected beam. NB: The strait-through passing beam does not experience a phase shift.
• recombine the two light beams in order to let them interfere.

Two types of beam splitters are commonly applied, the plate beam splitter and the cube beam splitter. According to some sources, these two different devices exhibit different phase shift characteristics, although the end result in the Mach-Zehnder is the same. For the understanding of the quantum implications of the behavior of these interferometers one should first try to understand how beam splitters function.

## Plate beam splitters

First we will look at the phase shifting behavior of the plate beam splitter as shown below.

When light travels from at less dense medium (air) and then reflects upon a denser medium (glass) it will experience a phase shift of 180o. The other way around, traveling from a more dense to a lesser dense medium, there is no phase shift. For more on phase shift of reflecting waves see WikiPedia. So the light traveling in the above figure from the right will, when reflected upwards, experience a 180o phase shift. Should it however travel from the left, go through the glass and then be reflected downwards, there will be no phase shift.

### Mach Zehnder with plate beam splitters

The figure below depict a Mach-Zehnder interferometer using plate beam splitters – 1 and 4 – and full mirrors – 2 and 3 – were the light reflects on silvered glass coming from air back into air.

We follow first the wave U traveling along the upper path 1-2-4 going to D1. Reflection at beam splitter 1 gives a 180o phase shift. Then another 180o phase shift happens at full mirror 2. There is no reflection at beam splitter 4 when going straight to D1. Total phase shift thus adds up to 360o. The wave D traveling along the lower path 1-3-4 going to D1 will experience 180o phase shift at mirror 3 and a 180o phase shift at beam splitter 4. Adds also up to 360o. So the phase difference in the recombined wave going from beam splitter 4 to D1 is zero. This means constructive interference, the waves add up. D1 receives light.

Now we follow the waves that should reach D2. The wave U traveling along 1-2-4 receives a 180o phase shift at beam splitter 1. Another 180o phase shift at full mirror 2. Then finally zero phase shift by the reflection from glass to air at beam splitter 4 in the direction of D2. Total is thus 360o. The wave D traveling along 1-3-4 receives only one phase shift at mirror 3 of 180o. So the recombined two waves going from beam splitter 4 to detector D2 will interfere with phase difference 180o, which means fully opposite phases. Which means destructive interference. D2 receives no light.

For a complete explanation which also covers the phase changes caused by traveling through the glass of beam splitters 1 and 4 here is a excellent webpage.

## Cube beam splitters

Cube beam splitters, as shown below, are constructed of two prisms with their oblique sides glued together with a special transparent resin. See figure below. Only one of these prisms is coated with a partial reflecting layer on the oblique side. Phase shift on reflection on that coating depends on the light having to travel before reflection through the resin or not. When interpreted in that way a Mach-Zehnder interferometer with cube beam splitters with the reflecting coating of the two cube beam splitters oriented in opposite ways will function exactly like the configuration with plate beam splitters.

However, some sources differ about the behavior of cube beam splitters.

This alternative behavior description for cube beam splitters says that, instead of the phase shift of 0o or 180o depending on the density of the medium behind the mirroring surface, like we saw with the plate beam splitter, that when the angle of reflection the incident beam makes with the reflecting surface is 45o, the reflected wave will experience a phase shift of 90o

### Mach-Zehnder with cube beam splitters

We will now consider the consequences of that alternative 90o phase shift on reflection interpretation for the interference at beam splitter 4 of the Mach-Zehnder configuration as shown below. We will see that the end result is the same. Detector D1 will receive all the light when upper and lower paths are equal in length..

Prisms 2 and 3 are full mirrors with the mirroring layer on their oblique side. Light is sent into these prisms perpendicular to the incoming surface, so there is no refraction, no change in direction of the beam. The beam will then hit the mirroring oblique back side with a 45o angle, reflect at 90o and leave the prism also perpendicular to the outgoing surface. With a reflection on the full mirror coating on the oblique back side of prisms 2 and 3 there occurs no phase shift because the incident beam arrives traveling in a more dense medium (glas) than that behind the mirror (air). Because of the lower light speed in the glass of those prisms the light will experience a slight extra phase shift going either path. These phase shifts will cancel each other out at beam splitter 4 when prism 2 and 3 are exactly the same. So we will ignore these phase shifts in the following explanation.

We follow first the wave traveling along the upper path 1-2-4 going to D1. Reflection at beam splitter 1 gives 90o phase shift. No phase shift at mirror 2. There is no reflection at beam splitter 4 when going straight to D1. Total phase shift is then 90o. The wave traveling along the lower path 1-3-4 going to D1 will experience no phase shift at beam splitter 1 and 3 and a phase shift of 90o at beam splitter 4. That also adds up to 90o. So the phase difference in the recombined waves going from 4 to detector D1 is zero. This means constructive interference, the recombined waves add up.

Now we follow the waves that should reach D2. The wave traveling along 1-2-4 receives a phase shift of 90o at reflection at beam splitter 1, then no phase shift at mirror 2 en finally a phase shift of 90o by the reflection at beam splitter 4 in the direction of D2. Total phase shift is 180o. The wave traveling along 1-3-4 experiences no phase shifts at all. Possibly check that for yourself. So the two combined waves going through beam splitter 4 in the direction of D2 will interfere with phase difference 180o, which means total opposite phases. Result is therefore destructive interference for the combined wave hitting D2.

This means that both phase shift interpretations result in the same interference behavior of the Mach-Zehnder interferometer.

## Open and closed configurations

The Mach-Zehnder interferometer described above contains always two full mirrors and two beam splitters. Such a configuration is called a ‘closed’ configuration. When the last beam splitter is removed, the configuration is called ‘open’. With a closed configuration and equal optical length from below and above, only detector 1 receives light. This is clearly the result of wave behavior. In an open configuration on the other hand, both detectors receive the same amount of light and wave behavior is now not evident. I advice that you do some thought exercise to try to explain this behavior yourself.

## EM-waves, photons and Quantum waves

All above explanations do use the image of light as EM-waves. Important however is here to note that the wave and phase shift approach for EM-waves can be applied in the same way to quantum state waves although these waves are not physical. Only when we try to explain the Mach-Zehnder with traveling photons that travel both ways at the same time we end up with problems and contradictions.

The destructive interference of the quantum wave hitting the D2 detector in a closed configuration means that the probability of the detection of a photon has become zero there. The constructive interference at the D1 detector means now that all photons will manifest there. Thus the quantum wave view predicts the same outcome as the EM-wave approach. Only when we start shooting individual photons we will still observe wavelike behavior resulting in every individual photon still registering on D1. It is then fully justified to ask how a single photon, which is considered to be a particle of pure energy, can travel both ways in order to interact with itself. Is it really appropriate to imagine a photon as a traveling particle of energy ignoring the confusion of that image?

The Mach-Zehnder interferometer is particularly suited for a quantum physics experiment that shows that quantum objects seem to adapt in “full flight” to ultra-fast changes in the measuring device. So fast that back-in-time effects seem to be occurring. These are the so-called delayed choice experiments.